Geometric Analysis and Surface Groups

Introduction

We propose to study links between curves in flag manifolds, surfaces solutions of geometric partial differential equations in some affine symmetric spaces, and functions on the moduli space of curves.

Energy Functions

We will consider the relevant energy functions on the moduli spaces of those curves, or on the moduli space of Anosov representations for periodic data, in particular in the context of positivity.

Goals

Amongst our concrete ambitious goals are:

1. Obtain topological invariants through quantizing Anosov deformation spaces.
2. Define and compute volumes of Anosov deformation spaces and prove recursion formulae for them.
3. Find surfaces in symmetric spaces associated to opers and the relevant higher-rank Liouville action.
4. Solve special cases of the Auslander conjecture using foliated spaces.

Project Backbone

More specifically, the backbone of this project is to explore a general class of functions on moduli spaces of Anosov representations and, beyond, of uniformly hyperbolic bundles.

Asymptotic Boundaries

Then, we propose to identify the family of curves that will be possible asymptotic boundaries -- in the spirit of quasisymmetric curves in the sphere -- the periodic ones corresponding to Anosov representations. We will prove the existence and uniqueness of surfaces bounded at infinity by these curves.

Area Consideration

Going back, we will consider the area of such a surface, both at critical points on the moduli space, and as a renormalizing function allowing to consider volumes of these moduli spaces.

Foliated Spaces

Finally, we will consider the space foliated by surfaces solutions of the asymptotic datum, and define entropy.